第 24833 題：有限群共軛類

本頁導覽

原題與已知條件
定義與適用範圍
逐步推導
逐步反向核對
結論
本題覆核與來源綁定

R0048-24833 · q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0033

原題與已知條件

由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-orbit-class；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

G001 — 由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-orbit-class；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

mko-finite-group-conjugacy-class

逐步推導

S001 · implies

操作前

source roles = [conjugacy-orbit-class]

操作後

derived = [[element, action_image_rows, conjugacy_class, certificate]]

條件：無

直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。

步驟識別與依賴

規則：source_ordered_fields_without_JS · 依賴：G001

S002 · implies

操作前

source roles = [Conjugacy ORIGINAL group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]

操作後

derived = [20, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19]], 是]

條件：無

先驗完整n×n閉合表與實際identity標籤，非預設0。mod44上限32，mod45上限64；同一carrier不能混入另一表。

步驟識別與依賴

規則：Group_complete_Cayley_signature_actual_identity · 依賴：G001, S001

S003 · implies

操作前

source roles = [Conjugacy ORIGINAL group, 20, FOR EVERY x,y,z in0..n-1]

操作後

derived = [[[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 8000, 是]

條件：無

矩陣(x,y)逐全部z核(xy)z=x(yz)，故涵蓋全部n³三元組，不是抽樣。表已在前步完整列出，不重印數百萬同義查表項。

步驟識別與依賴

規則：Group_ALL_associativity_triples_compact_complete_pair_matrix · 依賴：G001, S002

S004 · implies

操作前

source roles = [Conjugacy ORIGINAL group, 0]

操作後

derived = [[0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [[0, 0, 0, 0], [1, 9, 0, 0], [2, 8, 0, 0], [3, 7, 0, 0], [4, 6, 0, 0], [5, 5, 0, 0], [6, 4, 0, 0], [7, 3, 0, 0], [8, 2, 0, 0], [9, 1, 0, 0], [10, 10, 0, 0], [11, 11, 0, 0], [12, 12, 0, 0], [13, 13, 0, 0], [14, 14, 0, 0], [15, 15, 0, 0], [16, 16, 0, 0], [17, 17, 0, 0], [18, 18, 0, 0], [19, 19, 0, 0]], 是]

條件：無

每個元素的左、右逆都要回到同一實際identity；結合律使兩側逆唯一。這才完成群前提。

步驟識別與依賴

規則：Group_ALL_two_sided_inverses · 依賴：G001, S003

S005 · implies

操作前

source roles = [0, 0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S004

S006 · implies

操作前

source roles = [1, 9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S005

S007 · implies

操作前

source roles = [2, 8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S006

S008 · implies

操作前

source roles = [3, 7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S007

S009 · implies

操作前

source roles = [4, 6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S008

S010 · implies

操作前

source roles = [5, 5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S009

S011 · implies

操作前

source roles = [6, 4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S010

S012 · implies

操作前

source roles = [7, 3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S011

S013 · implies

操作前

source roles = [8, 2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S012

S014 · implies

操作前

source roles = [9, 1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S013

S015 · implies

操作前

source roles = [10, 10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S014

S016 · implies

操作前

source roles = [11, 11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S015

S017 · implies

操作前

source roles = [12, 12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S016

S018 · implies

操作前

source roles = [13, 13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S017

S019 · implies

操作前

source roles = [14, 14, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S018

S020 · implies

操作前

source roles = [15, 15, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S019

S021 · implies

操作前

source roles = [16, 16, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S020

S022 · implies

操作前

source roles = [17, 17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S021

S023 · implies

操作前

source roles = [18, 18, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S022

S024 · implies

操作前

source roles = [19, 19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

操作後

derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19]]

條件：無

對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。

步驟識別與依賴

規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S023

S025 · implies

操作前

source roles = [0, 20, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19]]]

操作後

derived = [是, [[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 是, (gh)x(gh)^-1=g(hxh^-1)g^-1]

條件：無

逐全部g,h,x核composition，亦由(gh)^-1=h^-1g^-1及結合律推同一公式；identity另驗。不是僅列出一個action表就假定axioms。

步驟識別與依賴

規則：Conjugacy_ALL_action_axioms_and_GENERAL_algebra · 依賴：G001, S024

S026 · implies

操作前

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19]]]

操作後

derived = [[[0], [1, 9], [10, 12, 14, 16, 18], [11, 13, 15, 17, 19], [2, 8], [3, 7], [4, 6], [5]], 20, 20, 是]

條件：無

e給自反、g^-1給反向、群乘積給傳遞；故所有完整orbits相等或不交並覆蓋G。按來源coset-like列表排序不代表它們是cosets；不引centralizer/class-size或類方程。

步驟識別與依賴

規則：Conjugacy_complete_orbits_equal_or_disjoint_partition · 依賴：G001, S025

S027 · implies

操作前

source roles = [1]

操作後

derived = [[[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 9], [11, 9], [12, 9], [13, 9], [14, 9], [15, 9], [16, 9], [17, 9], [18, 9], [19, 9]], [1, 9]]

條件：無

保每個acting g的像，再去重成排序的完整class。

步驟識別與依賴

規則：Conjugacy_actual_element_ALL_images_not_coset · 依賴：G001, S026

逐步反向核對

S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 9], [11, 9], [12, 9], [13, 9], [14, 9], [15, 9], [16, 9], [17, 9], [18, 9], [19, 9]], [1, 9], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。

來源雜湊與覆核憑證

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 3b31aa596a4ea9cc984161e370247d5a016254677ad17901f606e114eb29aa26
candidate record SHA-256: e7a89dd2a40971d8c721c237f59665aa93e8048b2536e1755124e0521c070dd4
semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON